To design a random effects one-way analysis of variance experiment one needs to determine both a, the number of treatments that are to be included in the experiment, as well asn, the number of times that each treatment is to be observed. In the fixed effects experiment only the number of times that each treatment is to be replicated needs to be determined since the number of treatments of interest is already fixed. We consider the problem of designing a random effects experiment where the total number of observations, N, is fixed and the power of theF-test is to be maximized. This problem is simple when one knows apriorithe value of the error standard deviation. We present a method which allows the experimenter to state a distribution for the error standard deviation. Our objective is then to find the experimental design which maximizes the expected power.